DLMF:10.43.E9 (Q3623)

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DLMF:10.43.E9
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    Statements

    test
    ∫ 0 x e ± t ⁢ t ν ⁢ K ν ⁡ ( t ) ⁢ d t = e ± x ⁢ x ν + 1 2 ⁢ ν + 1 ⁢ ( K ν ⁡ ( x ) ± K ν + 1 ⁡ ( x ) ) ∓ 2 ν ⁢ Γ ⁡ ( ν + 1 ) 2 ⁢ ν + 1 , superscript subscript 0 𝑥 superscript 𝑒 plus-or-minus 𝑡 superscript 𝑡 𝜈 modified-Bessel-second-kind 𝜈 𝑡 𝑡 minus-or-plus superscript 𝑒 plus-or-minus 𝑥 superscript 𝑥 𝜈 1 2 𝜈 1 plus-or-minus modified-Bessel-second-kind 𝜈 𝑥 modified-Bessel-second-kind 𝜈 1 𝑥 superscript 2 𝜈 Euler-Gamma 𝜈 1 2 𝜈 1 {\displaystyle{\displaystyle\int_{0}^{x}e^{\pm t}t^{\nu}K_{\nu}\left(t\right)% \mathrm{d}t=\frac{e^{\pm x}x^{\nu+1}}{2\nu+1}(K_{\nu}\left(x\right)\pm K_{\nu+% 1}\left(x\right))\mp\frac{2^{\nu}\Gamma\left(\nu+1\right)}{2\nu+1},}}
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    DLMF id
    DLMF:10.43.E9
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    constraint
    ℜ ⁡ ν > - 1 2 𝜈 1 2 {\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}
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